# Lagrange duality. The Lagrangian. We consider an optimization program of the form

Size: px
Start display at page:

Download "Lagrange duality. The Lagrangian. We consider an optimization program of the form"

## Transcription

1 Lagrange duality Another way to arrive at the KKT conditions, and one which gives us some insight on solving constrained optimization problems, is through the Lagrange dual. The dual is a maximization program in λ, ν it is always concave (even when the original program is not convex), and gives us a systematic way to lower bound the optimal value. The Lagrangian We consider an optimization program of the form minimize f 0 (x) f m (x) 0, m = 1,..., M (1) x R N h p (x) = 0, p = 1,..., P. Much of what we will say below applies equally well to nonconvex programs as well as convex programs, so we will make it clear when we are taking the f m to be convex and the h p to be affine. We will take the domain of all of the f m and h p to be all of R N below; this just simplifies the exposition, we can easily replace this with the intersections of the dom f m and dom h p. We will assume that the intersection of the feasible set, C = {x : f m (x) 0, h p (x) = 0, m = 1,..., M, p = 1,..., P } is a non-empty and a subset R N. 1

2 The Lagrangian takes the constraints in the program above and integrates them into the objective function. The Lagrangian L : R N R M R P R associated with this optimization program is L(x, λ, ν) = f 0 (x) + λ m f m (x) + ν p h p (x) The x above are referred to as primal variables, and the λ, ν as either dual variables or Lagrange multipliers. The Lagrange dual function g(λ, ν) : R M R P R is the minimum of the Lagrangian over all values of x: ( ) g(λ, ν) = inf f 0 (x) + λ m f m (x) + ν p h p (x). x R N Since the dual is a pointwise infimum of a family of affine functions in λ, ν, g is concave regardless of whether or not the f m, h p are convex. The key fact about the dual function is that is it is everywhere a lower bound on the optimal value of the original program. If p is the optimal value for (1), then g(λ, ν) p, for all λ 0, ν R P. 2

3 This is (almost too) easy to see. For any feasible point x 0, λ m f m (x 0 ) + ν p h p (x 0 ) 0, and so L(x 0, λ, ν) f 0 (x 0 ), for all λ 0, ν R P, meaning g(λ, ν) = inf x R N L(x, λ, ν) L(x 0, λ, ν) f 0 (x 0 ). Since this holds for all feasible x 0, g(λ, ν) inf x C f 0 (x) = p. The (Lagrange) dual to the optimization program (1) is maximize g(λ, ν) subject to λ 0. (2) λ R M, ν R P The dual optimal value d is d = sup g(λ, ν) = sup λ 0, ν λ 0, ν inf L(x, λ, ν). x R N Since g(λ, ν) p, we know that d p. The quantity p d is called the duality gap. If p = d, then we say that (1) and (2) exhibit strong duality. 3

4 Certificates of (sub)optimality Any dual feasible 1 (λ, ν) gives us a lower bound on p, since g(λ, ν) p. If we have a primal feasible x, then we know that f 0 (x) p f 0 (x) g(λ, ν). We will refer to f 0 (x) g(λ, ν) as the duality gap for primal/dual (feasible) pair x, λ, ν. We know that p [g(λ, ν), f 0 (x)], and likewise d [g(λ, ν), f 0 (x)]. If we are ever able to reduce this gap to zero, then we know that x is primal optimal, and λ, ν are dual optimal. There are certain kinds of primal-dual algorithms that produce a series of (feasible) points x (k), λ (k), ν (k) at every iteration. We can then use f 0 (x (k) ) g(λ (k), ν (k) ) ɛ, as a stopping criteria, and know that our answer would yield an objective value no further than ɛ from optimal. Strong duality and the KKT conditions Suppose that for a convex program, the primal optimal value p an the dual optimal value d are equal p = d. 1 We simply need λ 0 for (λ, ν) to be dual feasible. 4

5 If x is a primal optimal point and λ, ν is a dual optimal point, then we must have f 0 (x ) = g(λ, ν ) ( = inf x R N f 0 (x ) + f 0 (x ). f 0 (x) + λ mf m (x) + λ mf m (x ) + ) νph p (x) νph p (x ) The last inequality follows from the fact that λ m 0 (dual feasibility), f m (x ) 0, and h p (x ) = 0 (primal feasibility). Since we started out and ended up with the same thing, all of the things above must be equal, and so λ mf m (x ) = 0, m = 1,..., M. Also, since we know x is a minimizer of L(x, λ, ν ) (second equality above), which is an unconstrained convex function (with λ, ν fixed), the gradient with respect to x must be zero: x L(x, λ, ν ) = f 0 (x )+ λ m f m (x )+ νp h p (x ) = 0. Thus strong duality immediately leads to the KKT conditions holding at the solution. Also, if you can find x, λ, ν that obey the KKT conditions, not only do you know that you have a primal optimal point on your hands, but also we have strong duality (and λ, ν are dual optimal). For if KKT holds, x L(x, λ, ν ) = 0, 5

6 meaning that x is a minimizer of L(x, λ, ν ), i.e. thus L(x, λ, ν ) L(x, λ, ν ), g(λ, ν ) = L(x, λ, ν ) = f 0 (x ) + λ mf m (x ) + = f 0 (x ), (by KKT), νph p (x ) and we have strong duality. The upshot of this is that the conditions for strong duality are essentially the same as those under which KKT is necessary. The program (1) and its dual (2) have strong duality if the f m are affine inequality constraints, or there is an x R N such that for all the f i which are not affine we have f i (x) < 0. 6

7 Examples 1. Inequality LP. Calculate the dual of minimize x, c subject to Ax b. x R N Answer: The Lagrangian is L(x, λ) = x, c + λ m ( x, a m b m ) = c T x λ T b + λ T Ax. This is a linear functional in x it is unbounded below unless c + A T λ = 0. Thus g(λ) = inf x ( ) c T x λ T b + λ T Ax = { λ, b, c + A T λ = 0, otherwise. So the Lagrange dual program is maximize λ, b subject to A T λ = c λ R M λ 0. 7

8 2. Standard form LP. Calculate the dual of minimize x R N x, c subject to Ax = b λ 0. 8

9 Least-squares. Calculate the dual of minimize x R N x 2 2 subject to Ax = b. Check that the duality gap is zero. Answer: maximize ν R M 1 4 νt AA T ν b T ν 9

10 3. Minimum norm. Calculate the dual of minimize x subject to Ax = b, x R N where is a general valid norm. Answer: Use f 0 (x) = x to ease notation below. We start with the Lagrangian: L(x, ν) = f 0 (x) + ν p ( x, a m b m ) = f 0 (x) ν, b + (A T ν) T x and so g(ν) = ν, b + inf x = ν, b sup x = ν, b f 0 ( A T ν), ( ) f 0 (x) + (A T ν) T x ( f 0 (x) (A T ν) T x ) where f 0 is the Fenchel dual of f 0 : f 0 (y) = sup( x, y f 0 (x)). x With f 0 =, we know already that so f 0 (y) = g(ν) = { 0, y 1,, otherwise, { ν, b, A T ν 1, otherwise. 10

11 Thus the dual program is maximize ν, b subject to A T ν 1, ν R P where is the dual norm of. 11

### 5. Duality. Lagrangian

5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

### Convex Optimization M2

Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization

### Convex Optimization Boyd & Vandenberghe. 5. Duality

5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

### Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization

Extreme Abridgment of Boyd and Vandenberghe s Convex Optimization Compiled by David Rosenberg Abstract Boyd and Vandenberghe s Convex Optimization book is very well-written and a pleasure to read. The

### Lecture: Duality.

Lecture: Duality http://bicmr.pku.edu.cn/~wenzw/opt-2016-fall.html Acknowledgement: this slides is based on Prof. Lieven Vandenberghe s lecture notes Introduction 2/35 Lagrange dual problem weak and strong

### 14. Duality. ˆ Upper and lower bounds. ˆ General duality. ˆ Constraint qualifications. ˆ Counterexample. ˆ Complementary slackness.

CS/ECE/ISyE 524 Introduction to Optimization Spring 2016 17 14. Duality ˆ Upper and lower bounds ˆ General duality ˆ Constraint qualifications ˆ Counterexample ˆ Complementary slackness ˆ Examples ˆ Sensitivity

### I.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010

I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec - Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0

### Lecture: Duality of LP, SOCP and SDP

1/33 Lecture: Duality of LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2017.html wenzw@pku.edu.cn Acknowledgement:

### Lagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)

Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual

### Convex Optimization & Lagrange Duality

Convex Optimization & Lagrange Duality Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Convex optimization Optimality condition Lagrange duality KKT

### ICS-E4030 Kernel Methods in Machine Learning

ICS-E4030 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 28. September, 2016 Juho Rousu 28. September, 2016 1 / 38 Convex optimization Convex optimisation This

### EE/AA 578, Univ of Washington, Fall Duality

7. Duality EE/AA 578, Univ of Washington, Fall 2016 Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized

### Constrained Optimization and Lagrangian Duality

CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may

### subject to (x 2)(x 4) u,

Exercises Basic definitions 5.1 A simple example. Consider the optimization problem with variable x R. minimize x 2 + 1 subject to (x 2)(x 4) 0, (a) Analysis of primal problem. Give the feasible set, the

### On the Method of Lagrange Multipliers

On the Method of Lagrange Multipliers Reza Nasiri Mahalati November 6, 2016 Most of what is in this note is taken from the Convex Optimization book by Stephen Boyd and Lieven Vandenberghe. This should

### Duality. Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities

Duality Lagrange dual problem weak and strong duality optimality conditions perturbation and sensitivity analysis generalized inequalities Lagrangian Consider the optimization problem in standard form

### Optimization for Communications and Networks. Poompat Saengudomlert. Session 4 Duality and Lagrange Multipliers

Optimization for Communications and Networks Poompat Saengudomlert Session 4 Duality and Lagrange Multipliers P Saengudomlert (2015) Optimization Session 4 1 / 14 24 Dual Problems Consider a primal convex

### Lagrangian Duality and Convex Optimization

Lagrangian Duality and Convex Optimization David Rosenberg New York University February 11, 2015 David Rosenberg (New York University) DS-GA 1003 February 11, 2015 1 / 24 Introduction Why Convex Optimization?

### Optimization for Machine Learning

Optimization for Machine Learning (Problems; Algorithms - A) SUVRIT SRA Massachusetts Institute of Technology PKU Summer School on Data Science (July 2017) Course materials http://suvrit.de/teaching.html

### Tutorial on Convex Optimization: Part II

Tutorial on Convex Optimization: Part II Dr. Khaled Ardah Communications Research Laboratory TU Ilmenau Dec. 18, 2018 Outline Convex Optimization Review Lagrangian Duality Applications Optimal Power Allocation

### CS-E4830 Kernel Methods in Machine Learning

CS-E4830 Kernel Methods in Machine Learning Lecture 3: Convex optimization and duality Juho Rousu 27. September, 2017 Juho Rousu 27. September, 2017 1 / 45 Convex optimization Convex optimisation This

### Lecture 18: Optimization Programming

Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming

### Convex Optimization and Modeling

Convex Optimization and Modeling Duality Theory and Optimality Conditions 5th lecture, 12.05.2010 Jun.-Prof. Matthias Hein Program of today/next lecture Lagrangian and duality: the Lagrangian the dual

### Shiqian Ma, MAT-258A: Numerical Optimization 1. Chapter 4. Subgradient

Shiqian Ma, MAT-258A: Numerical Optimization 1 Chapter 4 Subgradient Shiqian Ma, MAT-258A: Numerical Optimization 2 4.1. Subgradients definition subgradient calculus duality and optimality conditions Shiqian

### Primal/Dual Decomposition Methods

Primal/Dual Decomposition Methods Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2018-19, HKUST, Hong Kong Outline of Lecture Subgradients

### EE364a Review Session 5

EE364a Review Session 5 EE364a Review announcements: homeworks 1 and 2 graded homework 4 solutions (check solution to additional problem 1) scpd phone-in office hours: tuesdays 6-7pm (650-723-1156) 1 Complementary

### Homework Set #6 - Solutions

EE 15 - Applications of Convex Optimization in Signal Processing and Communications Dr Andre Tkacenko JPL Third Term 11-1 Homework Set #6 - Solutions 1 a The feasible set is the interval [ 4] The unique

### Interior Point Algorithms for Constrained Convex Optimization

Interior Point Algorithms for Constrained Convex Optimization Chee Wei Tan CS 8292 : Advanced Topics in Convex Optimization and its Applications Fall 2010 Outline Inequality constrained minimization problems

### Lagrangian Duality Theory

Lagrangian Duality Theory Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapter 14.1-4 1 Recall Primal and Dual

### Lecture 7: Weak Duality

EE 227A: Conve Optimization and Applications February 7, 2012 Lecture 7: Weak Duality Lecturer: Laurent El Ghaoui 7.1 Lagrange Dual problem 7.1.1 Primal problem In this section, we consider a possibly

### Convex Optimization and SVM

Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence

### The Lagrangian L : R d R m R r R is an (easier to optimize) lower bound on the original problem:

HT05: SC4 Statistical Data Mining and Machine Learning Dino Sejdinovic Department of Statistics Oxford Convex Optimization and slides based on Arthur Gretton s Advanced Topics in Machine Learning course

### EE 227A: Convex Optimization and Applications October 14, 2008

EE 227A: Convex Optimization and Applications October 14, 2008 Lecture 13: SDP Duality Lecturer: Laurent El Ghaoui Reading assignment: Chapter 5 of BV. 13.1 Direct approach 13.1.1 Primal problem Consider

### CSCI : Optimization and Control of Networks. Review on Convex Optimization

CSCI7000-016: Optimization and Control of Networks Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one

### A Brief Review on Convex Optimization

A Brief Review on Convex Optimization 1 Convex set S R n is convex if x,y S, λ,µ 0, λ+µ = 1 λx+µy S geometrically: x,y S line segment through x,y S examples (one convex, two nonconvex sets): A Brief Review

### Convex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014

Convex Optimization Dani Yogatama School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA February 12, 2014 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12,

### Motivation. Lecture 2 Topics from Optimization and Duality. network utility maximization (NUM) problem:

CDS270 Maryam Fazel Lecture 2 Topics from Optimization and Duality Motivation network utility maximization (NUM) problem: consider a network with S sources (users), each sending one flow at rate x s, through

### 12. Interior-point methods

12. Interior-point methods Convex Optimization Boyd & Vandenberghe inequality constrained minimization logarithmic barrier function and central path barrier method feasibility and phase I methods complexity

### HW1 solutions. 1. α Ef(x) β, where Ef(x) is the expected value of f(x), i.e., Ef(x) = n. i=1 p if(a i ). (The function f : R R is given.

HW1 solutions Exercise 1 (Some sets of probability distributions.) Let x be a real-valued random variable with Prob(x = a i ) = p i, i = 1,..., n, where a 1 < a 2 < < a n. Of course p R n lies in the standard

### Applications of Linear Programming

Applications of Linear Programming lecturer: András London University of Szeged Institute of Informatics Department of Computational Optimization Lecture 9 Non-linear programming In case of LP, the goal

### Lagrange Relaxation and Duality

Lagrange Relaxation and Duality As we have already known, constrained optimization problems are harder to solve than unconstrained problems. By relaxation we can solve a more difficult problem by a simpler

### Constrained optimization

Constrained optimization DS-GA 1013 / MATH-GA 2824 Optimization-based Data Analysis http://www.cims.nyu.edu/~cfgranda/pages/obda_fall17/index.html Carlos Fernandez-Granda Compressed sensing Convex constrained

### ECE Optimization for wireless networks Final. minimize f o (x) s.t. Ax = b,

ECE 788 - Optimization for wireless networks Final Please provide clear and complete answers. PART I: Questions - Q.. Discuss an iterative algorithm that converges to the solution of the problem minimize

### Convex Optimization. Newton s method. ENSAE: Optimisation 1/44

Convex Optimization Newton s method ENSAE: Optimisation 1/44 Unconstrained minimization minimize f(x) f convex, twice continuously differentiable (hence dom f open) we assume optimal value p = inf x f(x)

### UC Berkeley Department of Electrical Engineering and Computer Science. EECS 227A Nonlinear and Convex Optimization. Solutions 5 Fall 2009

UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Solutions 5 Fall 2009 Reading: Boyd and Vandenberghe, Chapter 5 Solution 5.1 Note that

### Convex Optimization. Lecture 12 - Equality Constrained Optimization. Instructor: Yuanzhang Xiao. Fall University of Hawaii at Manoa

Convex Optimization Lecture 12 - Equality Constrained Optimization Instructor: Yuanzhang Xiao University of Hawaii at Manoa Fall 2017 1 / 19 Today s Lecture 1 Basic Concepts 2 for Equality Constrained

### Linear and Combinatorial Optimization

Linear and Combinatorial Optimization The dual of an LP-problem. Connections between primal and dual. Duality theorems and complementary slack. Philipp Birken (Ctr. for the Math. Sc.) Lecture 3: Duality

### Subgradient. Acknowledgement: this slides is based on Prof. Lieven Vandenberghes lecture notes. definition. subgradient calculus

1/41 Subgradient Acknowledgement: this slides is based on Prof. Lieven Vandenberghes lecture notes definition subgradient calculus duality and optimality conditions directional derivative Basic inequality

### Numerical Optimization

Linear Programming - Interior Point Methods Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Example 1 Computational Complexity of Simplex Algorithm

### A Tutorial on Convex Optimization II: Duality and Interior Point Methods

A Tutorial on Convex Optimization II: Duality and Interior Point Methods Haitham Hindi Palo Alto Research Center (PARC), Palo Alto, California 94304 email: hhindi@parc.com Abstract In recent years, convex

### Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem

Lecture 3: Lagrangian duality and algorithms for the Lagrangian dual problem Michael Patriksson 0-0 The Relaxation Theorem 1 Problem: find f := infimum f(x), x subject to x S, (1a) (1b) where f : R n R

### Lagrange Relaxation: Introduction and Applications

1 / 23 Lagrange Relaxation: Introduction and Applications Operations Research Anthony Papavasiliou 2 / 23 Contents 1 Context 2 Applications Application in Stochastic Programming Unit Commitment 3 / 23

### 4. Algebra and Duality

4-1 Algebra and Duality P. Parrilo and S. Lall, CDC 2003 2003.12.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone

### minimize x subject to (x 2)(x 4) u,

Math 6366/6367: Optimization and Variational Methods Sample Preliminary Exam Questions 1. Suppose that f : [, L] R is a C 2 -function with f () on (, L) and that you have explicit formulae for

### 10 Numerical methods for constrained problems

10 Numerical methods for constrained problems min s.t. f(x) h(x) = 0 (l), g(x) 0 (m), x X The algorithms can be roughly divided the following way: ˆ primal methods: find descent direction keeping inside

### Introduction to Mathematical Programming IE406. Lecture 10. Dr. Ted Ralphs

Introduction to Mathematical Programming IE406 Lecture 10 Dr. Ted Ralphs IE406 Lecture 10 1 Reading for This Lecture Bertsimas 4.1-4.3 IE406 Lecture 10 2 Duality Theory: Motivation Consider the following

### Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation

Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Instructor: Moritz Hardt Email: hardt+ee227c@berkeley.edu Graduate Instructor: Max Simchowitz Email: msimchow+ee227c@berkeley.edu

### The Karush-Kuhn-Tucker (KKT) conditions

The Karush-Kuhn-Tucker (KKT) conditions In this section, we will give a set of sufficient (and at most times necessary) conditions for a x to be the solution of a given convex optimization problem. These

### Lecture 9 Sequential unconstrained minimization

S. Boyd EE364 Lecture 9 Sequential unconstrained minimization brief history of SUMT & IP methods logarithmic barrier function central path UMT & SUMT complexity analysis feasibility phase generalized inequalities

### Tutorial on Convex Optimization for Engineers Part II

Tutorial on Convex Optimization for Engineers Part II M.Sc. Jens Steinwandt Communications Research Laboratory Ilmenau University of Technology PO Box 100565 D-98684 Ilmenau, Germany jens.steinwandt@tu-ilmenau.de

### Convex Optimization and Support Vector Machine

Convex Optimization and Support Vector Machine Problem 0. Consider a two-class classification problem. The training data is L n = {(x 1, t 1 ),..., (x n, t n )}, where each t i { 1, 1} and x i R p. We

### Lagrangian Duality. Richard Lusby. Department of Management Engineering Technical University of Denmark

Lagrangian Duality Richard Lusby Department of Management Engineering Technical University of Denmark Today s Topics (jg Lagrange Multipliers Lagrangian Relaxation Lagrangian Duality R Lusby (42111) Lagrangian

### Optimality Conditions for Constrained Optimization

72 CHAPTER 7 Optimality Conditions for Constrained Optimization 1. First Order Conditions In this section we consider first order optimality conditions for the constrained problem P : minimize f 0 (x)

### Support Vector Machines

Support Vector Machines Support vector machines (SVMs) are one of the central concepts in all of machine learning. They are simply a combination of two ideas: linear classification via maximum (or optimal

### Lecture 7: Convex Optimizations

Lecture 7: Convex Optimizations Radu Balan, David Levermore March 29, 2018 Convex Sets. Convex Functions A set S R n is called a convex set if for any points x, y S the line segment [x, y] := {tx + (1

### Quiz Discussion. IE417: Nonlinear Programming: Lecture 12. Motivation. Why do we care? Jeff Linderoth. 16th March 2006

Quiz Discussion IE417: Nonlinear Programming: Lecture 12 Jeff Linderoth Department of Industrial and Systems Engineering Lehigh University 16th March 2006 Motivation Why do we care? We are interested in

Additional Homework Problems Robert M. Freund April, 2004 2004 Massachusetts Institute of Technology. 1 2 1 Exercises 1. Let IR n + denote the nonnegative orthant, namely IR + n = {x IR n x j ( ) 0,j =1,...,n}.

### Lecture 8. Strong Duality Results. September 22, 2008

Strong Duality Results September 22, 2008 Outline Lecture 8 Slater Condition and its Variations Convex Objective with Linear Inequality Constraints Quadratic Objective over Quadratic Constraints Representation

### Example Problem. Linear Program (standard form) CSCI5654 (Linear Programming, Fall 2013) Lecture-7. Duality

CSCI5654 (Linear Programming, Fall 013) Lecture-7 Duality Lecture 7 Slide# 1 Lecture 7 Slide# Linear Program (standard form) Example Problem maximize c 1 x 1 + + c n x n s.t. a j1 x 1 + + a jn x n b j

### Nonlinear Optimization: What s important?

Nonlinear Optimization: What s important? Julian Hall 10th May 2012 Convexity: convex problems A local minimizer is a global minimizer A solution of f (x) = 0 (stationary point) is a minimizer A global

### Duality. Geoff Gordon & Ryan Tibshirani Optimization /

Duality Geoff Gordon & Ryan Tibshirani Optimization 10-725 / 36-725 1 Duality in linear programs Suppose we want to find lower bound on the optimal value in our convex problem, B min x C f(x) E.g., consider

### Convex Optimization. Ofer Meshi. Lecture 6: Lower Bounds Constrained Optimization

Convex Optimization Ofer Meshi Lecture 6: Lower Bounds Constrained Optimization Lower Bounds Some upper bounds: #iter μ 2 M #iter 2 M #iter L L μ 2 Oracle/ops GD κ log 1/ε M x # ε L # x # L # ε # με f

### Introduction to Machine Learning Lecture 7. Mehryar Mohri Courant Institute and Google Research

Introduction to Machine Learning Lecture 7 Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu Convex Optimization Differentiation Definition: let f : X R N R be a differentiable function,

### Lecture 6: Conic Optimization September 8

IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions

### Numerical Optimization

Constrained Optimization Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Constrained Optimization Constrained Optimization Problem: min h j (x) 0,

### Primal-dual Subgradient Method for Convex Problems with Functional Constraints

Primal-dual Subgradient Method for Convex Problems with Functional Constraints Yurii Nesterov, CORE/INMA (UCL) Workshop on embedded optimization EMBOPT2014 September 9, 2014 (Lucca) Yu. Nesterov Primal-dual

### Solution to EE 617 Mid-Term Exam, Fall November 2, 2017

Solution to EE 67 Mid-erm Exam, Fall 207 November 2, 207 EE 67 Solution to Mid-erm Exam - Page 2 of 2 November 2, 207 (4 points) Convex sets (a) (2 points) Consider the set { } a R k p(0) =, p(t) for t

### 4TE3/6TE3. Algorithms for. Continuous Optimization

4TE3/6TE3 Algorithms for Continuous Optimization (Duality in Nonlinear Optimization ) Tamás TERLAKY Computing and Software McMaster University Hamilton, January 2004 terlaky@mcmaster.ca Tel: 27780 Optimality

### Support Vector Machines: Maximum Margin Classifiers

Support Vector Machines: Maximum Margin Classifiers Machine Learning and Pattern Recognition: September 16, 2008 Piotr Mirowski Based on slides by Sumit Chopra and Fu-Jie Huang 1 Outline What is behind

### 10-725/ Optimization Midterm Exam

10-725/36-725 Optimization Midterm Exam November 6, 2012 NAME: ANDREW ID: Instructions: This exam is 1hr 20mins long Except for a single two-sided sheet of notes, no other material or discussion is permitted

### Equality constrained minimization

Chapter 10 Equality constrained minimization 10.1 Equality constrained minimization problems In this chapter we describe methods for solving a convex optimization problem with equality constraints, minimize

### Generalization to inequality constrained problem. Maximize

Lecture 11. 26 September 2006 Review of Lecture #10: Second order optimality conditions necessary condition, sufficient condition. If the necessary condition is violated the point cannot be a local minimum

### Part IB Optimisation

Part IB Optimisation Theorems Based on lectures by F. A. Fischer Notes taken by Dexter Chua Easter 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after

### Optimality, Duality, Complementarity for Constrained Optimization

Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of Wisconsin-Madison May 2014 Wright (UW-Madison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear

### LECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE

LECTURE 25: REVIEW/EPILOGUE LECTURE OUTLINE CONVEX ANALYSIS AND DUALITY Basic concepts of convex analysis Basic concepts of convex optimization Geometric duality framework - MC/MC Constrained optimization

### Lecture Note 5: Semidefinite Programming for Stability Analysis

ECE7850: Hybrid Systems:Theory and Applications Lecture Note 5: Semidefinite Programming for Stability Analysis Wei Zhang Assistant Professor Department of Electrical and Computer Engineering Ohio State

### SECTION C: CONTINUOUS OPTIMISATION LECTURE 11: THE METHOD OF LAGRANGE MULTIPLIERS

SECTION C: CONTINUOUS OPTIMISATION LECTURE : THE METHOD OF LAGRANGE MULTIPLIERS HONOUR SCHOOL OF MATHEMATICS OXFORD UNIVERSITY HILARY TERM 005 DR RAPHAEL HAUSER. Examples. In this lecture we will take

### Karush-Kuhn-Tucker Conditions. Lecturer: Ryan Tibshirani Convex Optimization /36-725

Karush-Kuhn-Tucker Conditions Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Given a minimization problem Last time: duality min x subject to f(x) h i (x) 0, i = 1,... m l j (x) = 0, j =

### E5295/5B5749 Convex optimization with engineering applications. Lecture 5. Convex programming and semidefinite programming

E5295/5B5749 Convex optimization with engineering applications Lecture 5 Convex programming and semidefinite programming A. Forsgren, KTH 1 Lecture 5 Convex optimization 2006/2007 Convex quadratic program

### CSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods

CSCI 1951-G Optimization Methods in Finance Part 09: Interior Point Methods March 23, 2018 1 / 35 This material is covered in S. Boyd, L. Vandenberge s book Convex Optimization https://web.stanford.edu/~boyd/cvxbook/.

### Convex Optimization Lecture 13

Convex Optimization Lecture 13 Today: Interior-Point (continued) Central Path method for SDP Feasibility and Phase I Methods From Central Path to Primal/Dual Central'Path'Log'Barrier'Method Init: Feasible&#

### Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method

Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method Robert M. Freund March, 2004 2004 Massachusetts Institute of Technology. The Problem The logarithmic barrier approach

### COM S 578X: Optimization for Machine Learning

COM S 578X: Optimization for Machine Learning Lecture Note 4: Duality Jia (Kevin) Liu Assistant Professor Department of Computer Science Iowa State University, Ames, Iowa, USA Fall 2018 JKL (CS@ISU) COM

### Example: feasibility. Interpretation as formal proof. Example: linear inequalities and Farkas lemma

4-1 Algebra and Duality P. Parrilo and S. Lall 2006.06.07.01 4. Algebra and Duality Example: non-convex polynomial optimization Weak duality and duality gap The dual is not intrinsic The cone of valid

### Duality Uses and Correspondences. Ryan Tibshirani Convex Optimization

Duality Uses and Correspondences Ryan Tibshirani Conve Optimization 10-725 Recall that for the problem Last time: KKT conditions subject to f() h i () 0, i = 1,... m l j () = 0, j = 1,... r the KKT conditions

### Duality in Linear Programs. Lecturer: Ryan Tibshirani Convex Optimization /36-725

Duality in Linear Programs Lecturer: Ryan Tibshirani Convex Optimization 10-725/36-725 1 Last time: proximal gradient descent Consider the problem x g(x) + h(x) with g, h convex, g differentiable, and

### Lecture 8 Plus properties, merit functions and gap functions. September 28, 2008

Lecture 8 Plus properties, merit functions and gap functions September 28, 2008 Outline Plus-properties and F-uniqueness Equation reformulations of VI/CPs Merit functions Gap merit functions FP-I book:

### Duality (Continued) min f ( x), X R R. Recall, the general primal problem is. The Lagrangian is a function. defined by

Duality (Continued) Recall, the general primal problem is min f ( x), xx g( x) 0 n m where X R, f : X R, g : XR ( X). he Lagrangian is a function L: XR R m defined by L( xλ, ) f ( x) λ g( x) Duality (Continued)